Ray's Arithmetic · Book 4

Ray's New Higher Arithmetic

The last and fullest book of Ray's arithmetic series, for students about 12 and up: number theory, fractions and decimals, the old and metric measures, ratio and proportion, percentage and its business uses, with definitions, rules, worked examples and thousands of problems. Every session has a guide you can follow step by step, with the right page of the book beside it.

0 of 127 lessons done (saved in this browser) Start Lesson 1 →

How to use it

  • One guide, one sitting. Each guide covers a few of the book's numbered articles and takes 30 to 45 minutes. A few long problem sets are split over two or three days; the guide says so.
  • Read the definitions aloud. Ray states every idea as a definition, principle or rule. Read it together, put it in plain words, then let the worked example show what it means.
  • Work the book's example first. Each guide walks through the example the book solves, then two real problems together, then a set your student works alone.
  • The book prints most answers. Cover them with a card while working, then check. Where the book prints none, or the scan is unreadable, the guide gives the answer.
  • Keep a notebook. A page of definitions and rules in your student's own words, and neat worked problems, make the best review.
  • Old money and measures. Mills, farthings, rods, pecks and troy ounces fill many problems. The guides explain them briefly; the arithmetic is what matters.

Unit 1 Introduction

Definitions of quantity, science, art and number, and the signs of arithmetic. Lessons 1–3.

  1. 1What arithmetic is aboutArts. 1–8✓
  2. 2Propositions, rules and numbersArts. 9–21✓
  3. 3The signs of arithmeticArts. 22–39✓

Unit 2 Numeration and Notation

Reading and writing whole numbers of any size, the old English method, and Roman numerals. Lessons 4–6.

  1. 4Orders and periodsArts. 40–47✓
  2. 5Reading and writing large numbersArt. 48✓
  3. 6The English method and Roman notationArts. 49–50✓

Unit 3 Addition

The meaning, principles, rule and proofs of addition, with practical problems. Lessons 7–8.

  1. 7Addition: definition and principlesArts. 51–52✓
  2. 8Adding: the rule, proofs and problemsArt. 52, Examples 1–19✓

Unit 4 Subtraction

Subtraction with its principles and proof, and a first look at accounts, debits and credits. Lessons 9–10.

  1. 9SubtractionArts. 53–54✓
  2. 10Business terms: accounts and balancesArts. 55–58✓

Unit 5 Multiplication

Short and long multiplication, bills and receipts, and the book's short cuts. Lessons 11–15.

  1. 11Multiplication: terms and the short multiplierArts. 59–62✓
  2. 12Long multiplication and practical problemsArt. 63✓
  3. 13Bills, receipts and statementsArts. 64–66✓
  4. 14Short cuts: composite multipliers and ciphersArts. 67–69✓
  5. 15More short cuts in multiplicationArts. 70–71✓

Unit 6 Division

Long and short division, the order of operations, general principles, short cuts and mixed problems. Lessons 16–23.

  1. 16Division: terms and principlesArts. 72–78✓
  2. 17Long divisionArt. 78, Long Division✓
  3. 18Short divisionArt. 78, Short Division✓
  4. 19Short cuts in divisionArts. 79–81✓
  5. 20Arithmetical signs and the order of operationsArts. 82–86✓
  6. 21General principles of multiplication and divisionArt. 87✓
  7. 22Short cuts with parts of 100Arts. 88–90✓
  8. 23General problems and miscellaneous exercisesGeneral Problems; Miscellaneous Exercises✓

Unit 7 Properties of Numbers

Primes, factoring, tests of divisibility, the greatest common divisor, least common multiple, casting out nines and cancellation. Lessons 24–29.

  1. 24Kinds of numbersArt. 91✓
  2. 25Factoring and tests of divisibilityArts. 92–94✓
  3. 26Common factors, all divisors and the greatest common divisorArts. 95–100✓
  4. 27The least common multipleArts. 101–104✓
  5. 28Casting out ninesArt. 105✓
  6. 29CancellationArt. 106✓

Unit 8 Common Fractions

What fractions are, how to reduce them, the four operations, and the G. C. D. and L. C. M. of fractions. Lessons 30–38.

  1. 30What a fraction isArts. 107–123✓
  2. 31Reading, writing and the principles of fractionsArts. 124–129✓
  3. 32Reduction of fractionsArts. 130–135✓
  4. 33Common denominatorsArts. 136–139✓
  5. 34Adding and subtracting fractionsArts. 140–141✓
  6. 35Multiplying fractionsArt. 142✓
  7. 36Dividing fractions and complex fractionsArts. 143–144✓
  8. 37G. C. D. and L. C. M. of fractionsArts. 145–146✓
  9. 38Promiscuous exercises in fractionsPromiscuous Exercises✓

Unit 9 Decimal Fractions

Reading and writing decimals, changing them to fractions and back, and the four operations. Lessons 39–43.

  1. 39Decimal fractions: notationArts. 147–156✓
  2. 40Changing decimals and common fractionsArts. 157–159✓
  3. 41Adding and subtracting decimalsArts. 160–161✓
  4. 42Multiplying decimalsArts. 162–163✓
  5. 43Dividing decimalsArts. 164–166✓

Unit 10 Circulating Decimals

Why some decimals repeat, and how to work with repeating decimals. Lessons 44–45.

  1. 44Circulating decimals: what they areArts. 167–171✓
  2. 45Working with circulatesArts. 172–177✓

Unit 11 Compound Denominate Numbers

The tables of money, weight, length, area, volume, capacity, angles, time and the metric system, and working with mixed units. Lessons 46–58.

  1. 46Measures, value and United States moneyArts. 178–181✓
  2. 47English, French and German moneyArts. 182–184✓
  3. 48Measures of weightArts. 185–190✓
  4. 49Measures of length and surfaceArts. 191–198✓
  5. 50Solids, capacity and anglesArts. 199–204✓
  6. 51Time, longitude and other tablesArts. 205–208✓
  7. 52The metric systemArts. 209–217✓
  8. 53Reduction of compound numbersArts. 218–220✓
  9. 54Reduction practiceArts. 219–220, Examples 1–56✓
  10. 55Adding and subtracting compound numbersArts. 221–222✓
  11. 56Multiplying and dividing compound numbersArts. 223–224✓
  12. 57Longitude and timeArt. 225✓
  13. 58Aliquot partsArt. 226✓

Unit 12 Ratio

Ratios, their terms and principles. Lessons 59–59.

  1. 59RatioArts. 227–229✓

Unit 13 Proportion

Simple and compound proportion, and the method of cause and effect. Lessons 60–62.

  1. 60Proportion and the rule of threeArts. 230–231✓
  2. 61Simple proportion: practiceArt. 231, Examples 1–49✓
  3. 62Compound proportionArts. 232–233✓

Unit 14 Percentage

Base, rate and percentage, and the four cases of percentage. Lessons 63–64.

  1. 63Percentage: base, rate and percentageArts. 234–237✓
  2. 64Finding the baseArts. 238–241✓

Unit 15 Percentage: Profit and Loss, Stocks and Bonds

The first business applications: profit and loss, and stocks, dividends and assessments. Lessons 65–67.

  1. 65Profit and loss: finding profit and rateArts. 242–245✓
  2. 66Profit and loss: finding the costArts. 246–247✓
  3. 67Stocks and bondsArts. 248–253✓

Unit 16 Percentage: Applications without Time (continued)

Premium and discount, commission, stock investments, insurance, taxes and customs duties, all worked as cases of percentage. Lessons 68–76.

  1. 68Premium and discount: words and Case IArts. 254–256✓
  2. 69Premium and discount: Cases II to IVArts. 257–259✓
  3. 70Commission and brokerage: words, Cases I and IIArts. 260–263✓
  4. 71Commission: Cases III and IVArts. 264–265✓
  5. 72Stock investments: words and Case IArts. 266–269✓
  6. 73Stock investments: Cases II to VArts. 270–273✓
  7. 74InsuranceArts. 274–278✓
  8. 75TaxesArts. 279–284✓
  9. 76United States revenue and customs dutiesArts. 285–290✓

Unit 17 Percentage with Time: Interest

Simple interest by several methods, its five cases, promissory notes and annual interest. Lessons 77–83.

  1. 77Interest: the wordsArts. 291–294✓
  2. 78Finding interest: the common method and aliquot partsArts. 295–296✓
  3. 79The six per cent method and exact interestArt. 297✓
  4. 80Finding the time and the rateArts. 298–299✓
  5. 81Finding the principalArts. 300–301✓
  6. 82Promissory notesArts. 302–303✓
  7. 83Annual interestArt. 304✓

Unit 18 Percentage with Time: Partial Payments

Settling a note paid off in parts, by the United States, state and mercantile rules. Lessons 84–85.

  1. 84Partial payments: the United States ruleArts. 305–307✓
  2. 85Connecticut, Vermont and mercantile rules optionalArts. 308–309✓

Unit 19 Percentage with Time: True and Bank Discount

Present worth, banks, checks and drafts, and how banks discounted notes in the 1880s. Lessons 86–89.

  1. 86True discount and present worthArts. 310–312✓
  2. 87Banks, checks and draftsArt. 313✓
  3. 88Bank discount and proceedsArts. 314–315✓
  4. 89Bank discount: face of a note, and rates of interestArts. 316–318✓

Unit 20 Percentage with Time: Exchange

Paying distant debts with drafts and bills, at home and abroad, and comparing routes through other cities. Lessons 90–92.

  1. 90Exchange and domestic exchangeArts. 319–320✓
  2. 91Foreign exchangeArt. 321✓
  3. 92Arbitration of exchange optionalArt. 322✓

Unit 21 Percentage with Time: Equation of Payments and Settlement of Accounts

The fair single date for several debts, account currents, account sales and storage. Lessons 93–95.

  1. 93Equation of paymentsArts. 323–326✓
  2. 94Settlement of accountsArts. 327–329✓
  3. 95Account sales and storage accounts optionalArts. 330–332✓

Unit 22 Percentage with Time: Compound Interest and Annuities

Compound interest by hand and by table, annuities, life annuities and life insurance. Lessons 96–101.

  1. 96Compound interestArts. 333–334✓
  2. 97Compound interest by tables, and Cases II to IVArts. 335–338✓
  3. 98Annuities and perpetuitiesArts. 339–341✓
  4. 99Annuities certain: present value, final value, payment, time and rateArts. 342–347✓
  5. 100Life annuities, dowers and reversions optionalArts. 348–353✓
  6. 101Life insurance optionalArts. 354–355✓

Unit 23 Partnership

Sharing gains and losses among partners, and dividing a bankrupt's property among creditors. Lessons 102–104.

  1. 102Partnership: shares for the same timeArts. 356–357✓
  2. 103Partnership: shares for different timesArt. 358✓
  3. 104BankruptcyArt. 359✓

Unit 24 Alligation

Averages of mixtures, and how much of each kind to mix for a wanted price. Lessons 105–107.

  1. 105Alligation medial: the average of a mixtureArts. 360–361✓
  2. 106Alligation alternate: how much of each?Arts. 362–363✓
  3. 107Alligation alternate: limited quantities and totalsArts. 364–365✓

Unit 25 Involution

Powers, exponents, and the square and cube of a sum. Lessons 108–109.

  1. 108PowersArts. 366–370✓
  2. 109Squaring and cubing a sumArts. 371–372✓

Unit 26 Evolution

Square, cube and higher roots, and their use with triangles and similar figures. Lessons 110–116.

  1. 110Roots, and why the square root method worksArts. 373–374✓
  2. 111Extracting the square rootArts. 375–377✓
  3. 112Extracting the cube rootArt. 378✓
  4. 113Cube roots of products and fractions; any rootArts. 379–382✓
  5. 114Horner's method optionalArts. 383–384✓
  6. 115Triangles and the right triangleArts. 385–388✓
  7. 116Similar figures and general exercises in rootsArt. 389✓

Unit 27 Series

Arithmetical and geometrical progressions. Lessons 117–118.

  1. 117Arithmetical progressionArts. 390–393✓
  2. 118Geometrical progressionArts. 394–396✓

Unit 28 Mensuration

Lines and figures, areas, the circle, solids, and practical measures of masonry, casks, lumber, grain and hay. Lessons 119–124.

  1. 119Lines, angles, surfaces and polygonsArts. 397–403✓
  2. 120Areas of triangles and quadrilateralsArt. 404✓
  3. 121Regular polygons and the circleArts. 405–409✓
  4. 122Solids: surfaces and volumesArt. 410✓
  5. 123Masonry, brickwork and gaugingArts. 411–415✓
  6. 124Lumber, grain and hayArts. 416–419✓

Unit 29 Miscellaneous Exercises

Mixed problems from the whole book, to finish. Lessons 125–127.

  1. 125Miscellaneous exercises, part 1Misc. Exercises 1–35✓
  2. 126Miscellaneous exercises, part 2Misc. Exercises 36–78✓
  3. 127Miscellaneous exercises, part 3Misc. Exercises 79–106✓

The book itself

The book is divided into numbered articles, not lessons. Each guide covers a few articles, shown beside its title; guide numbers simply count the sessions in order. The guides are written for this site; the book is in the public domain.

@teddyrookbook GitHub